KEY STAGE Mark scheme 3 - Emaths - Free … KS3 SA… · Tiers 3–5, 4–6, ... 2008 KS3 Mathematics test mark scheme: Paper 2 Introduction Introduction ... out the given unit, accept

Mathematics tests

Mark scheme for Paper 1Tiers 3–5, 4–6, 5–7 and 6–8

National curriculum assessments

2008

ALL TIERS

Ma

KEY STAGE

3

Mathematics tests

Mark scheme for Paper 2Tiers 3–5, 4–6, 5–7 and 6–8

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22

2008 KS3 Mathematics test mark scheme: Paper 2 Introduction

Introduction

The test papers will be marked by external markers. The markers will follow the mark scheme in this booklet, which is provided here to inform teachers.

This booklet contains the mark scheme for paper 2 at all tiers. The paper 1mark scheme is printed in a separate booklet. Questions have been given namesso that each one has a unique identifi er irrespective of tier.

The structure of the mark schemes

The marking information for questions is set out in the form of tables, which start on page 11 of this booklet. The columns on the left-hand side of each table provide a quick reference to the tier, question number, question part and the total number of marks available for that question part.

The Correct response column usually includes two types of information:

a statement of the requirements for the award of each mark, with an indication of whether credit can be given for correct working, and whether the marks are independent or cumulative

examples of some different types of correct response, including the most common.

The Additional guidance column indicates alternative acceptable responses, and provides details of specifi c types of response that are unacceptable. Other guidance, such as when ‘follow-through’ is allowed, is provided as necessary.

Questions with a UAM element are identifi ed in the mark scheme by an encircled U with a number that indicates the signifi cance of using and applying mathematics in answering the question. The U number can be any whole number from 1 to the number of marks in the question.

For graphical and diagrammatic responses, including those in which judgements on accuracy are required, marking overlays have been provided as the centre pages of this booklet.

The 2008 key stage 3 mathematics tests and mark schemes were developed by the Test Development Team at Edexcel.

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33

2008 KS3 Mathematics test mark scheme: Paper 2 General guidance

General guidance

Using the mark schemes

Answers that are numerically equivalent or algebraically equivalent are acceptable unless the mark scheme states otherwise.

In order to ensure consistency of marking, the most frequent procedural queries are listed on the following two pages with the prescribed correct action. This is followed by further guidance relating specifi cally to the marking of questions that involve money, negative numbers, algebra, time, coordinates or probability. Unless otherwise specifi ed in the mark scheme, markers should apply the following guidelines in all cases.

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4


What if …

The pupil’s responsedoes not match

closely any of the examples given.

Markers should use their judgement in deciding whether the response corresponds with the statement of requirements given in the Correct response column. Refer also to the Additional guidance.

The pupil hasresponded in a

non-standard way.

Calculations, formulae and written responses do not have to be set out in any particular format. Pupils may provide evidence in any form as long as its meaning can be understood. Diagrams, symbols or words are acceptable for explanations or for indicating a response. Any correct method of setting out working, however idiosyncratic, is acceptable. Provided there is no ambiguity, condone the continental practice of using a comma for a decimal point.

The pupil has made a conceptual error.

In some questions, a method mark is available provided the pupil has madea computational, rather than conceptual, error. A computational error isa ‘slip’ such as writing 4 × 6 = 18 in an otherwise correct long multiplication. A conceptual error is a more serious misunderstanding of the relevant mathematics; when such an error is seen, no method marks may be awarded. Examples of conceptual errors are: misunderstanding of place value, such as multiplying by 2 rather than 20 when calculating 35 × 27; subtracting the smaller value from the larger in calculations such as 45 – 26 to give the answer 21; incorrect signs when working with negative numbers.

The pupil’s accuracyis marginal

according to the overlay provided.

Overlays can never be 100% accurate. However, provided the answer is within, or touches, the boundaries given, the mark(s) should be awarded.

The pupil’s answer correctly follows

through from earlier incorrect work.

Follow-through marks may be awarded only when specifi cally stated in the mark scheme, but should not be allowed if the diffi culty level of the question has been lowered. Either the correct response or an acceptable follow-through response should be marked as correct.

There appears to be a misreading affecting

the working.

This is when the pupil misreads the information given in the question anduses different information. If the original intention or diffi culty level of the question is not reduced, deduct one mark only. If the original intention or diffi culty level is reduced, do not award any marks for the question part.

The correct answer is in the wrong place.

Where a pupil has shown understanding of the question, the mark(s) should be given. In particular, where a word or number response is expected, a pupil may meet the requirement by annotating a graph or labelling a diagram elsewhere in the question.

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5


What if … Marking procedure

The fi nal answer iswrong but the

correct answer is shown in the

working.

Where appropriate, detailed guidance will be given in the mark scheme and must be adhered to. If no guidance is given, markers will need to examine each case to decide whether:

the incorrect answer is due to a transcription error If so, award the mark.

in questions not testing accuracy, the correct answer has been given but then rounded or truncated

If so, award the mark.

the pupil has continued to give redundant extra working which does not contradict work already done

If so, award the mark.

the pupil has continued, in the same part of the question, to give redundant extra working which does contradict work already done.

If so, do not award the mark. Where a question part carries more than one mark, only the fi nal mark should be withheld.

The pupil’s answer is correct but the wrong

working is seen.

A correct response should always be marked as correct unless the markscheme states otherwise.

The correct responsehas been crossed

or rubbed outand not replaced.

Mark, according to the mark scheme, any legible crossed or rubbed outwork that has not been replaced.

More than one answer is given.

If all answers given are correct or a range of answers is given, all of which are correct, the mark should be awarded unless prohibited by the mark scheme.If both correct and incorrect responses are given, no mark should be awarded.

The answer is correctbut, in a later part of

the question, the pupil has

contradicted thisresponse.

A mark given for one part should not be disallowed for working or answers given in a different part, unless the mark scheme specifi cally states otherwise.

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6


Marking specifi c types of question

Responses involving moneyFor example: £3.20 £7

Accept Do not accept

Any unambiguous indication of the correct amount

eg £3.20(p), £3 20, £3,20,3 pounds 20, £3-20,£3 20 pence, £3:20, £7.00

The unit, £ or p, is usually printed in the answer space. Where the pupil writes an answer outside the answer space with no units, accept responses that are unambiguous when considered alongside the given units

eg with £ given in the answer space, accept 3.207 or 7.00

Given units amendedeg with £ crossed out in the

answer space, accept 320p700p

Incorrect or ambiguous indication of the amount

eg £320, £320p or £700p

Ambiguous use of units outside the answer space

eg with £ given in the answerspace, do not accept3.20p outside the answerspace

Incorrect placement of decimal points, spaces, etc or incorrect use or omission of 0

eg £3.2, £3 200, £32 0, £3-2-0,£7.0

Responses involving negative numbersFor example: –2


To avoid penalising the error below more than once within each question, do not award the mark for the fi rst occurrence of the error within each question. Where a question part carries more than one mark, only the fi nal mark should be withheld.

Incorrect notationeg 2 –

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7


Responses involving the use of algebraFor example: 2 + n n + 2 2n n

2 n2

Accept Take care ! Do not accept

Unambiguous use of a different caseor variable

eg N used for nx used for n

Words used to precede or follow equations or expressions

eg t = n + 2 tiles ortiles = t = n + 2for t = n + 2

! Unconventional notationeg n × 2 or 2 × n or n2

or n + n for 2nn × n for n2

n ÷ 2 for n2

or 12

n

2 + 1n for 2 + n2 + 0n for 2

Within a question that demands simplifi cation, do not accept as part of a fi nal answer involving algebra.Accept within a method when awarding partial credit, or within an explanation or general working.

Embedded values given when solving equations

eg in solving 3x + 2 = 32,3 × 10 + 2 = 32 for x = 10

To avoid penalising the two types of error below more than once within each question, do not award the mark for the fi rst occurrence of each type within each question. Where a question part carries more than one mark, only the fi nal mark should be withheld.

! Words or units used within equations or expressions

eg n tiles + 2n cm + 2

Do not accept on their own.Ignore if accompanying an acceptable response.

Unambiguous letters used to indicate expressions

eg t = n + 2 for n + 2

Ambiguous letters used to indicate expressions

eg n = n + 2 for n + 2

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8


Responses involving timeA time interval For example: 2 hours 30 minutes


Any unambiguous indicationeg 2.5 (hours), 2h 30

Digital electronic timeie 2:30

Incorrect or ambiguous time intervaleg 2.3(h), 2.30, 2-30, 2h 3,

2.30min

! The unit, hours and/or minutes, is usually printed in the answer space.Where the pupil writes an answer outside the answer space, or crosses out the given unit, accept answers with correct units, unless the question has specifi cally asked for other units to be used.

A specifi c time For example: 8:40am 17:20


Any unambiguous, correct indicationeg 08.40, 8.40, 8:40, 0840, 8 40,

8-40, twenty to nine, 8,40

Unambiguous change to 12 or 24 hour clock

eg 17:20 as 5:20pm, 17:20pm

Incorrect timeeg 8.4am, 8.40pm

Incorrect placement of separators, spaces, etc or incorrect use or omission of 0

eg 840, 8:4:0, 084, 84

Responses involving coordinatesFor example: ( 5, 7 )


Unconventional notationeg ( 05, 07 )

( fi ve, seven )

( x5,

y7 )

( x = 5, y = 7 )

Incorrect or ambiguous notationeg ( 7, 5 )

( y7,

x5 )

( 5x, 7y )( 5x, 7y )( x – 5, y – 7 )

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9


Responses involving probabilityA numerical probability should be expressed as a decimal, fraction orpercentage only.

For example: 0.7 7

10 70%


Equivalent decimals, fractions and percentages

eg 0.700, 70100 ,

3550 , 70.0%

The fi rst four categories of error below should be ignored if accompanied by an acceptable response, but should not be accepted on their own.However, to avoid penalising the fi rst three types of error below more than once within each question, do not award the mark for the fi rst occurrence of each type of errorunaccompanied by an acceptable response. Where a question part carries more than one mark, only the fi nal mark should be withheld.

! A probability that is incorrectly expressed

eg 7 in 107 over 107 out of 107 from 10

! A probability expressed as a percentage without a percentage sign.

! A fraction with other than integers in the numerator and/or denominator.

! A probability expressed as a ratioeg 7 : 10, 7 : 3, 7 to 10

A probability correctly expressed in one acceptable form which is then incorrectly converted, but is still less than 1 and greater than 0

eg 70100 =

1825

A probability greater than 1 or less than 0

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1010


Recording marks awarded on the test paper

All questions, even those not attempted by the pupil, will be marked, with a 1 or a 0 entered in each marking space. Where 2m can be split into 1m gained and 1m lost, with no explicit order, then this will be recorded by the marker as 1

0The total marks awarded for a double page will be written in the box at the bottom of the right-hand page, and the total number of marks obtained on the paper will be recorded on the front of the test paper.

A total of 120 marks is available in each of tiers 3–5, 4–6, 5–7 and 6–8.

Awarding levels

The sum of the marks gained on paper 1, paper 2 and the mental mathematicspaper determines the level awarded. Level threshold tables, which show the mark ranges for the award of different levels, will be available on the NAAwebsite www.naa.org.uk/tests from Monday 23 June 2008.

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11

2008 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 only

Tier & Question Rounding3-5 4-6 5-7 6-8

1 Correct response Additional guidance

a 2m

or

Matches all four numbers correctly, ie

912

990

955

849

881

1000

900

800

! Number matched to more than onenearest hundredFor 2m or 1m, do not accept as a correct match

1m Matches at least two numbers correctly

b 1m Gives a value greater than or equal to 45 but less than 55

Fractions or decimals

Value of exactly 55 given

1m

U1

Gives a different value greater than or equal to 45 but less than 55 from any credited for the fi rst mark

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12


Tier & Question Cuboid3-5 4-6 5-7 6-8


a 1m 6

b 1m 2

1m 3

Tier & Question Placing 403-5 4-6 5-7 6-8


1m Indicates 40 in the correct position, ie

0 200

! Inaccurate indicationAccept provided their indication is closer to the correct marker than any other

1m Indicates 40 in the correct position, ie

0 400

! Follow-throughFor the second mark, accept responses in which the distance between the arrow and zero is half as big as for the fi rst mark

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13


Tier & Question Directions3-5 4-6 5-7 6-8


a 1m Indicates right then left Unambiguous indicationeg, for part (a)

r then l

b 2m

or

Gives directions that state or imply the following four steps (or equivalent) in the correct order:

1. (Come out of house A and) turn right2. (Take the) second road on the left3. Turn right4. (House C is on the) right

For part (b), unambiguous description for step 2, ie ‘second road on the left’eg

Cross the junction then turn left At the next turning, go straight on, then turn left

1m

U1

Gives directions that state or imply all four steps, with not more than one erroreg

RightLeft [indication of ‘second’ omitted]RightRight

Turn right out of the houseTake the second right (error)Take the fi rst rightThe house is on the right

or

Gives directions that state or imply steps 2 and 3 above, even if steps 1 and/or 4 are incorrect or omitted

or

Gives correct directions for getting from house C to house A:

1. (Come out of house C and) turn left2. (At the end of the road) turn left3. Turn right4. (House A is on the) left

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14


Tier & Question Theme park3-5 4-6 5-7 6-8


a 1m 8

b 1m 7

c 1m 5

Tier & Question Writing cheques3-5 4-6 5-7 6-8


1m £ 102.70 ! Non-standard notationCondone any unambiguous notationeg, for the fi rst mark accept

£ 102 = 701m £ 120.07

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15


Tier & Question Adding odd3-5 4-6 5-7 6-8


1m

U1

Gives a correct counter example showing the sum of two odd numberseg

1 + 3 = 4, which is even 5 and 7 makes 12 Odd = even + 1, so

odd + odd = even + even + 2= even

Minimally acceptable exampleeg

1 + 3 = 4

Odd numbers taken to be equaleg

2 × 5 = 10

! Response uses negative numbers and/or zeroAccept negative odd numbers and zero as an even number within a correct responseeg, accept

–1 + 1 = 0

! Other calculations or general reasoning given alongside a correct responseIgnore other calculations, even if they are incorrect or do not relate to the given statementIf a correct counter example is given, ignore any general explanation unless it contradicts the counter example given

Incomplete or incorrect exampleeg

1 + 3 = even Odd + odd = even Only odd + even = odd 15 + 17 = 42

Tier & Question Calculating3-5 4-6 5-7 6-8


a 1m 2134

b 1m 663768

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16

2008 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6

Tier & Question Four cards3-5 4-6 5-7 6-8

10 2 Correct response Additional guidance

2m

or

Matches all four statements to their correct positions, ie

… odd number

… the number 2

… less than 7

… 4 times table Certain

Even chance

Impossible

… even number

! Statement matched to more than one positionFor 2m or 1m, do not accept as a correct match

1m Matches any two statements to their correct positions

Tier & Question Time machine3-5 4-6 5-7 6-8


2m

or

6

1m Shows the value 94 or the values 4 and 2

or

Shows a complete correct method with not more than one computational erroreg

100 – 46 – 48 100 – (46 + 48) 100 – 46 = 53 (error)

53 – 48 = 5

For 1m, necessary brackets omittedeg

100 – 46 + 48

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17


Tier & Question Sleep3-5 4-6 5-7 6-8


a a 1m 11 –11

b b 2m

or

7pm or 19:00 ! Incorrect notation for timeCondoneeg, for 2m accept

7 19

7am

1m

U1

Shows or implies that 12 hours’ sleep are neededeg

12 seen (30 – 6) ÷ 2 30 – 6 = 24, 24 ÷ 2

! For 1m, necessary brackets omittedCondoneeg, for 1m accept

30 – 6 ÷ 2

! For 1m, incorrect order of operations shownCondone provided evaluation using the correct order is seeneg, for 1m accept

6 – 30 = 24, 24 ÷ 2eg, for 1m do not accept

(6 – 30) ÷ 2

For 1m, –12

Tier & Question Sorting shapes3-5 4-6 5-7 6-8


2m

or

Gives the three letters B, C and D in the correct places in the table, ie

A D

B C

Unambiguous indication

Any letter repeated in an incorrect place inthe tableeg, for 1 mark

A A (error) D

B C

eg, for 0 marks

A A (error) D

C (error) B C

1m Gives at least two of the letters in the correct places in the table, with not more than one erroror omission

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18


Tier & Question Shopping3-5 4-6 5-7 6-8


a a 1m £ 1.15

b b 1m 5 ! Reference to remainderCondone reference to the correct amount of money left overeg, accept

5 with 20p change 5 r 20

eg, do not accept 5.5(…) or 5.6 5 with 55p change

Tier & Question Speedometer3-5 4-6 5-7 6-8


a a 1m Indicates the correct value on the scale, ie

40

20 80

1000

60

mph

! Inaccurate indicationAccept provided their marker would touch the circumference of the dial within 2mm of the correct position, if extended

b b 1m 40

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19


Tier & Question Jug3-5 4-6 5-7 6-8


1m 750

1m 100

1m 15 or equivalent fraction or decimal

Tier & Question Football survey3-5 4-6 5-7 6-8


2m

or

Gives the value 3 in the keyandcompletes 3 circles for each of the Yes and No rows

! Circles not shaded, or inaccurate in sizeAccept provided the pupil’s intention is clear

1m

U1

Shows or implies the value 9eg

Completes 9 circles for one or both rows

or

Draws the same number of circles for each of the Yes and No rows, provided this number is not 4, even if the value in the key is incorrect or omitted

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20


Tier & Question Double shape3-5 4-6 5-7 6-8


a a 1m Indicates Yesandgives a correct explanation

The most common correct explanations:

Show or imply the correct areaseg

The area of the rectangle is 18, the area of the square is 9 and 9 × 2 = 18

A is 18 and B is 18 ÷ 2 = 9

Refer to the space taken up by each shapeeg

Two of the squares can fi t inside the rectangle

If you draw a line down the middle of the rectangle, you get two of the squares

A holds twice as many squares as B

Refer to the ratio of lengths together with the equal widthseg

They are the same width but the rectangle is twice as long as the square

6 × 3 is twice 3 × 3

! Incorrect unitsCondoneeg, accept

18cm, 9cm 182, 92

Minimally acceptable explanationeg

18, 9 2 × 9 (or double 9), 9 18, 18 ÷ 2

Incomplete explanationeg

The area of the rectangle is 18


A holds two squares You cut A in half to get B Rectangle divided into two squares on the diagram

I counted the squares inside the shapes


The area of A is twice the area of B B is half of A He’s just added another shape on I counted the squares


Equal width, but the length is doubled Same height, but width is twice as long 6 × 3, 3 × 3


The rectangle is twice as long as the square

Because A is 6 squares long and B is 3 squares long

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21


Tier & Question Double shape (cont)3-5 4-6 5-7 6-8


b b 1m

U1

Indicates Noandgives a correct explanation


Show or imply the correct perimeterseg

The perimeter of the rectangle is 18, the perimeter of the square is 12 but 2 × 12 18

2 × 9 is not twice 2 × 6

Refer to the distance around each shapeeg

The length around the edge of the square goes more than halfway round the edge of the rectangle

Refer to the rectangle’s additional lengthseg

You only add two of the square’s sides to get the rectangle, not all four

It’s increased by 50%, not doubled You join two squares, but two of their sides

will be touching

! Incorrect unitsCondoneeg, accept

18cm2, 12cm2


18, 12 2 × 9, 2 × 6 12 + 6, 12 It’s 6cm more but that’s not double 12

Incomplete or incorrect explanationeg

The perimeter of the rectangle is 18 Area A = 18, area B = 12


It’s less than double the perimeter of the square

B’s perimeter is more than half A’s I counted the distance round the sides


The perimeter of A is not double the perimeter of B


It has two extra lengths of 3, not four It’s half as long again

These sidesare hidden


It has two extra sides

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22

2008 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7

Tier & Question Track3-5 4-6 5-7 6-8

19 11 2 Correct response Additional guidance

a a a 1m 5 ! Response assumes the piece of track shown has already been countedFor answers of 4 for part (a) followed by5 for part (b), mark as 0, 1

b b b 1m 6

Tier & Question Cube edges3-5 4-6 5-7 6-8


2m

or

Completes the table correctly to show the further 5 ways with no errors or duplicates,ie

Ways of moving from A to H

A B C H

A B G H

A D C H

A D E H

A F E H

A F G H

[rows in any order]

Unambiguous indicationeg, for A B G H

ABGH

! Correct vertices, but in an incorrect ordereg, for A B G H

A G B HDo not accept as a correct way

1m Gives at least 3 of the correct ways, even if there are other errors or duplicates

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23


Tier & Question Matching expressions3-5 4-6 5-7 6-8


2m

or

Matches all four statements correctly, ie

2

2 – a

a + 2

2a

a – 2

2a

2a

a2

! Statement matched to more than one expressionFor 2m or 1m, do not accept as a correct match

1m Matches three of the statements correctly

Tier & Question Area3-5 4-6 5-7 6-8


1m Gives both correct areas, ie 9 then 3

Tier & Question Values3-5 4-6 5-7 6-8


a a a 1m 6 ! Incomplete processingPenalise only the fi rst occurrenceeg, for parts (a) and (b)

9 – 3 4 – 6

Mark as 0, 1b b b 1m –2

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24


Tier & Question Symmetry patterns3-5 4-6 5-7 6-8


a a a 1m Indicates two squares so that the shape has rotation symmetry of order 4, ie

Unambiguous indication

b b b 1m Indicates four squares in total [that include the same two squares required in part (a)] so that the shape has rotation symmetry of order 2eg

! For part (b), response uses part squaresAccept provided the intended symmetry is clearly correct

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25


Tier & Question Shop3-5 4-6 5-7 6-8


2m

or

£ 196.25

1m

U1

Digits 19625 seen

or

Shows or implies the correct subtotals of pay for the hours worked at 6.35, or pay for thehours worked at 7.5(0)eg

158.75 25 × 6.35 37.5(0) 5 × 7.5(0) 15 and 22.5(0) seen

or

Shows the values 44.45, 40.4(0) and 22.5(0)

or

Shows or implies a complete correct method with not more than one computational erroreg

7 + 7 + 4 + 7 = 26 (error), 26 × 6.35 + 5 × 7.5(0) = 202.60

or

Gives an answer of 193.95 or 200.85[the only error is to assume 6.35 or 7.50 for all hours on Wednesday]

Markers may fi nd the following useful:

Mon 7 × 6.35 = 44.45Tues 7 × 6.35 = 44.45Wed 4 × 6.35 = 25.4 and 2 × 7.5 = 15 or 4 × 6.35 + 2 × 7.5 = 40.4(0) Thur 7 × 6.35 = 44.45(Fri 0)Sat 3 × 7.5 = 22.5

no. of hoursworked

pay per hour total

25 6.35 158.755 7.5(0) 37.5(0)

or2 7.5(0) 15(.00)3 7.5(0) 22.5(0)

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26


Tier & Question Using algebra3-5 4-6 5-7 6-8


1m n + 2 ! Unsimplifi ed expression or unconventional notationeg, for Jo’s age

n + 1 + 1 1n + 2

eg, for Kate’s age 2 × (n + 2) n × 2 + 4

Condone

1m 2(n + 2) or 2n + 4 ! For the second mark, follow-throughAccept follow-through as 2 × their algebraic expression for Jo’s age provided there are no other errorseg, from Jo’s age as 2n accept

4n n × 4

For the second mark, incomplete processingeg

2 × n + 2 × 2

For the second mark, necessary brackets omittedeg

2 × n + 2 2(n + 2

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27


Tier & Question Goldbach3-5 4-6 5-7 6-8


a a a 1m Gives a pair of prime numbers that sum to 16,ie

3 and 13, in either orderor

5 and 11, in either order

1m

U1

Gives a different pair of prime numbers that sum to 16 from any credited for the fi rst mark

Values credited for the fi rst mark repeated but in reverse order

b b b 1m

U1

Completes the sentence correctly, giving an even number greater than 16 and a correct pair of prime numbers that sum to their numbereg

… even number 20 … … prime numbers 7 and 13 … even number 22 …

… prime numbers 11 and 11 … even number 50 …

… prime numbers 3 and 47

Their even number is less than or equalto 16

Markers may fi nd the following values useful:

Prime numbers up to 100

2, 3, 5, 7

11, 13, 17, 19

23, 29

31, 37

41, 43, 47

53, 59

61, 67

71, 73, 79

83, 89

97

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28


Tier & Question Side length3-5 4-6 5-7 6-8


2m

or

6.3 or equivalent

1m

U1

Shows the value 25.2 or equivalent

or

Shows a complete correct method with not more than one computational erroreg

8.4 × 3 ÷ 4 (8.4 + 8.4 + 8.4) ÷ 4 8.4 + 8.4 + 8.4 = 25.6 (error),

25.6 ÷ 4 = 6.4

For 1m, necessary brackets omittedeg

8.4 + 8.4 + 8.4 ÷ 4

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29


Tier & Question

Value of x3-5 4-6 5-7 6-8


a a a 1m

U1

Indicates … one particular number, ie

b b b 1m

U1

Indicates … any number at all, ie

Tier & Question Darts3-5 4-6 5-7 6-8


1m Gives all three correct numbers, ie

10, 15 and 20 [any order]

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30


Tier & Question Conversions3-5 4-6 5-7 6-8


1m

U1

Gives a correct explanation


Show the values in grams do not consistently go up/down in steps of 25 per ounceeg

It goes up in 25s until the step from 3 to 4 ounces when it suddenly goes up 35

It should go from 150g down to 125g, but it’s 110g instead

Show that the relationship between twovalues in grams is not what other valueswould predicteg

If 1 ounce is 25g, then 4 ounces should be 25 × 4 = 100g not 110g

If 5 ounces is 150g, then 10 ounces should be 150 × 2 = 300g not 275g

10 ounces in grams should be25 × 10 = 250, but it is 275 in the table

50 ÷ 2 = 25, but 150 ÷ 5 = 30

Explanation does not use the values in the given tableeg

1 ounce is more like 28g They only use 25g as roughly equal, so those values are not accurate

! Explanation states or implies what values ‘should be’ or that the table is ‘incorrect’Condone


It goes up in 25s at fi rst but then changes

It goes up 25, 25, 35, 40 and so it is not a steady pattern

It should go 25, 50, 75, 100 The numbers should go up by the same amount each time


25, 25, 35, 40 4 ounces should be 100g and

10 ounces should be 250g They don’t go up in proportion


25 × 4 110 4 should be 25 × 4 = 100 150 × 2 275 If 5 is 150, then 10 should be 300 50 ÷ 2 150 ÷ 5 10oz should equal double 5oz but it doesn’t


1 ounce is 25g so 4 ounces shouldn’t be 110g

5 ounces = 150g, but 10 ounces = 275g

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31


Tier & Question Concorde3-5 4-6 5-7 6-8


2m

or

1200

1m Shows or implies a correct rate, other than1 mile every 3 seconds, even if it doesn’t use single units of timeeg

20 (miles) per minute

13

(mile) in a sec

10 miles in 30 seconds 60 miles every 3 mins

or

Shows or implies a complete correct method with not more than one computational or rounding erroreg

20 × 60

603

× 60

13 × 3600

1 ÷ 3 = 0.33 (premature rounding)0.33 × 602 = 1188

! For 1m, unit(s) abbreviatedCondone provided unambiguous within the context of the questioneg, for 1m accept

20m per min

13

m/s [miles implied by given context]

eg, for 1m do not accept 20m per m [ambiguity between miles and minutes]

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32


Tier & Question Counters in a bag3-5 4-6 5-7 6-8


2m

or

Completes the sentence correctly with three positive integers r, w then y, such thatw = 2r and y < reg

2, 4 then 1 3, 6 then 1 or 2 4, 8 then 1, 2 or 3

1m Completes the sentence with three integersr, w then y, such that w = 2r and y = 0eg

2, 4 then 0 3, 6 then 0

or

Completes the sentence with three values r, w then y between zero and one, such that

r > 14

, w = 2r and r + w + y = 1

eg

27

, 47

then 17

0.3, 0.6 then 0.1

For 1m, values for r or w negative or zeroeg

–1, –2 then 0 0, 0 then 0

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33


Tier & Question Perimeters3-5 4-6 5-7 6-8


a a a 1m 7a + 3 ! Unsimplifi ed expression or unconventional notationeg

42a + 186

(42 × a + 18) ÷ 6Condone

Necessary brackets omittedeg

42a + 18 ÷ 6

b b b 1m 5

c c c 1m 24 ! Units givenIgnore, even if incorrect for a perimetereg, accept

24cm 24cm2

Incomplete processingeg

4 × 6

Tier & Question Yoghurt3-5 4-6 5-7 6-8


2m

or

125

1m Shows or implies recognition of the need to divide by 7eg

57

× 175

175 ÷ 7 25 seen

or

Shows the value 50 [mass of fruit]

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34


Tier & Question Lawn3-5 4-6 5-7 6-8


2m

or

28.(…) or 9π

1m Shows or implies a complete correct method for fi nding the area of the lawn, with no evidence of conceptual error and not more than one computational or rounding erroreg

Shows the digits 282(…) or 283 32 × π π = 3 (rounding error), 9 × 3 = 27

For 1m, conceptual erroreg

32 × π = 19 or 18.8(…) or 6π π32 = 89 Area = 2 × 3 × π

Tier & Question Triangular numbers3-5 4-6 5-7 6-8


a a a 1m 55

b b b 1m 5050

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35


Tier & Question Isosceles triangle3-5 4-6 5-7 6-8


2m

or

Gives x = 74, y = 32 and z = 46andgives a correct reason for each angle

The most common correct reasons:

For angle x, refer to the isosceles triangleeg

It is an isosceles triangle, so it is equal to angle ADB

The triangle is isosceles so it is the same as the 74° angle marked

For angle y, refer to angles in a triangleeg

Angles in a triangle, so 180 – 74 – 74 74 + 74 = 148 and 180 – 148 because they

add up to 180 in a triangle

For angle z, refer to angles in a triangle and angles on a straight line or just angles in a triangle or exterior angle of a triangleeg

Angles in a triangle, 180 – 28 – 74 – 32 Angles on a straight line, 180 – 74 = 106,

angles in a triangle, 180 – 106 – 28 Exterior angle of a triangle, 74 – 28

Minimally acceptable reasoneg

Isosceles

Incomplete reason without the correct geometrical property identifi edeg

It is equal to angle ADB It is the same as the 74° angle marked


Angles in a triangle

! Follow-through from their xFor angle y, accept 106 – their x accompanied by a correct reason


180 – 74 – 74 74 + 74 = 148 and 180 – 148


Angles in a triangle Angles on a straight line and anglesin a triangle

Exterior angle of a triangle

! Follow-through from their x and their yFor angle z, accept 152 – their x – their y accompanied by a correct reason


180 – 28 – 74 – 32 180 – 74 = 106, 180 – 106 – 28

1m

U1

Gives two correct angles with a correct reason for each

or

Gives all three correct angles, even if reasons are incorrect or omitted

For 1m, follow-throughAccept follow-through for each angle as detailed above

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36


Tier & Question Journeys3-5 4-6 5-7 6-8


a a 1m Gives all four names in the correct order, ie

ChrisDeeAnnBen

Unambiguous indicationeg

CDAB

b b 2m

or

Joins the points (0, 0), (15, 1), (30, 1.5) and (60, 4) with straight lines, ie

0 0

1

2

3

4

10 20 30 40 50 60

! Lines not ruled or accurateAccept provided the pupil’s intention is clear

1m Indicates at least two of the points (15, 1),(30, 1.5) and (60, 4) on the graph, even if they are not joined or are joined incorrectly

or

Shows or implies all three sets of coordinates (15, 1), (30, 1.5) and (60, 4) in working, even if the graph is incorrect or omitted

! For 1m, follow-through from their (15, 1) with an incorrect y-valueFor an incorrect y-value between 0.5 and 3 inclusive, accept their (30, 1.5) as(30, their incorrect y-value + 0.5)eg, for 1m accept

0 0

1

2

3

4

10 20 30 40 50 60

c c 1m 5 ! Follow-through from their graph in part (b)Provided their line for the fi nal section of the graph has a positive gradient and passes through (60, 4), accept follow-through as2 × (4 – their y-coordinate for (30, 1.5))

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37


Tier & Question Special offer3-5 4-6 5-7 6-8


2m

or

Indicates Both paid the sameandgives a correct justifi cationeg

Marie paid 96 – 9.60 = 86.40Richard paid 108 – 21.60 = 86.40

0.9 × 96 = 86.40.8 × 108 = 86.4

For 2m, minimally acceptable justifi cationeg

96 – 9.6(0), 108 – 21.6(0) 0.9 × 96, 0.8 × 108 86.4(0)

For 2m or 1m, incomplete justifi cationeg

10% off 96 is the same as 20% off 108 It works out to be the same

1m

U1

Gives a correct justifi cation but makes an incorrect or no decision

or

Gives a correct justifi cation with not more than one computational or rounding error, but follows through to make their correct decisioneg

Marie paid 96 – 9.60 = 87.4(0) (error)Richard paid 108 – 21.60 = 86.4(0)

[indicates Marie]

For 1m, conceptual erroreg

20% off 108 = 108 – (108 ÷ 20) = 108 – 5.40 = 102.60

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38


Tier & QuestionMarking overlay available Planes

3-5 4-6 5-7 6-8


a a 1m Indicates the correlation is positive ! Positive qualifi edIgnoreeg, accept

Strong positive Direct positive

Sign of correlation not indicatedeg

High Strong

! Relationship quantifi edIgnore alongside a correct response

Relationship described without referenceto correlationeg

The greater the wingspan, the more passengers it can hold

b b 1m Draws a line of best fi t within the tolerance, and at least of the length, as shown on the overlay

! Line not ruled or accurateAccept provided the line is within tolerance, and at least of the length required

! Line of best fi t is incorrect beyond the dashed lines on the overlayCondoneeg, accept

A correct line of best fi t that is then joined to the origin

c c 2m

or

3600 to 5200 inclusive

1m

U1

Shows a value between 180 and 260 inclusive

or

Shows a value that follows through from their line of best fi teg

Their line passes through the point (40, 280), fi nal answer: 5600

! For 1m, range for follow-through valueIf their line goes through (40, y) accept follow-through as 20 × (y ± 10) provided their line always has a positive gradient

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39


Tier & Question Cubes3-5 4-6 5-7 6-8


2m

or

27

1m Shows the values 216 (or 63 or 6 × 6 × 6) and8 (or 23 or 2 × 2 × 2), even if there are errors

or

Shows or implies that 3 of the smaller cubes will fi t along each edge of the larger cubeeg

33 or 3 × 3 × 3 3 by 3 by 3

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40


Tier & Question Best buy3-5 4-6 5-7 6-8


2m

or

Indicates Aandgives a correct justifi cation, based on correctly calculating a pair of comparable values

The most common justifi cations:

Compare pence (or pounds) per grameg

159 ÷ 454 = 0.35(…) 125 ÷ 340 = 0.36(…) (or 0.37)

Compare grams per penny (or per pound)eg

454 ÷ 159 = 2.8(…) (or 2.9)340 ÷ 125 = 2.7(2)

454 ÷ 1.59 = 285.(…) (or 286)340 ÷ 1.25 = 272

Reason proportionally using the priceseg

125 ÷ 340 × 454 = 166.(…) (or 167)That’s more than 159

159 ÷ 454 × 340 = 119.(…), which is < 125 1.59 × 340 = 540(.6) (or 541)

1.25 × 454 = 567(.5) (or 568) 2 × 340 = 680g, which is £2.50

1.5 × 454 = 681g, which is only £2.39 4 × 340g = 1360g for £5

3 × 454g = 1362g for £4.77 If A were decreased by 114g its price should

go down by 40p (or 39.(..)p), but the difference is 34p so it’s a worse reduction

454 – 340 = 114g, £1.59 – £1.25 = 34p

but 114340

× 1.25 = 42p (or 41.(…)p)

For 2m, correct decision and any pair of comparable values shownNote that common pairs are:0.35(…) and 0.36(…) or 0.37 (p per g)0.0035(…) and 0.0036(…) or 0.0037 (£ per g)2.8(…) or 2.9 and 2.7(2) (g per p)285.(…) or 286 and 272 (g per £)159 and 166.(…) or 167 (p per 454g)119.(…) and 125 (p per 340g)540(.6) or 541 and 567(.5) or 568 (£ per 154 360g)34 and 39.(…) or 40 (p for 114g extra compared to A)34 and 41.(…) or 42 (p for 114g extra compared to B)

! Correct decision and comparison is per 454g or per 340g but given price is not restatedCondoneeg, for 2m accept

125 ÷ 340 × 454 = 167

! Correct decision but units omitted, incorrect or inconsistentCondone provided any values used to make a decision are comparableeg, for 2m accept

1.59 ÷ 454 = 0.351.25 ÷ 340 = 0.37

! Additional incorrect workingIgnore

For 2m or 1m, incomplete justifi cationeg

454 – 340 = 114g£1.59 – £1.25 = 34pTherefore jar A because you get 114g more for only 34p extra

For 2m or 1m, comparable values, or the method to calculate them, not showneg

The big jar is 8p cheaper

1m

U1

Shows a correct pair of comparable values but makes an incorrect or no decision

or

Shows correct calculations for a pair of comparable values, with not more than one error if evaluation is attempted, then follows through to make their correct decisioneg

159 ÷ 454 and 125 ÷ 340, so A 454 ÷ 159 = 2.8(…)

340 ÷ 125 = 27.2 (error), so B

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41


Tier & Question Shadows3-5 4-6 5-7 6-8


2m

or

4.2 or equivalent

1m Shows the value 23

or 32

or equivalents

or

Shows or implies a complete correct method with not more than one computational or rounding erroreg

1.8 ÷ 2.7 × 6.3 1.8 ÷ 2.7 = 0.6 (rounding error)

0.6 × 6.3 = 3.78 6.3 ÷ 2.7 = 2.3 (rounding error)

1.8 × 2.3 = 4.14

! For 1m, value rounded

For 23

, accept 0.66(…) or 0.67

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42


Tier & Question 1, 2, 43-5 4-6 5-7 6-8


3m

or

Gives a complete correct response that satisfi es all four of the following conditions:

1. Indicates that A is 82. Indicates that B is 73. Indicates that C is 84. Shows or implies correct substitution at

least for value Ceg

4(42 – 3 × 4 + 8)6

4 × 126

48 ÷ 6

2m

or

Gives a response that satisfi es three of the four conditions

1m Gives a response that satisfi es two of the four conditions

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43


Tier & Question Triangles3-5 4-6 5-7 6-8


a 2m

or

14.4(…), or 4 13, or 208 ! Value of 14Do not accept unless a correct method or a more accurate value is seen

For 2m or 1m, method uses accurate orscale drawing

1m Shows a correct method that indicates at least the intention to square and subtract the two given lengthseg

172 – 92

289 – 81 208 seen

b 2m

or

7.8 or 7.79(…) ! Value of 8Do not accept unless a correct method or a more accurate value is seen

For 2m or 1m, method uses accurate or scale drawing

1m Shows or implies a correct trigonometric ratio involving not more than one unknowneg

Answer of 7.7 12 tan 33

tan 33 = DF12

tan 33 = 0.6 (premature rounding),12 × 0.6 = 7.2

tan 57 = 12x

! For 1m, no indication of which angle is being consideredeg

tan = DF12

For 1m, accept only if the trigonometric ratio is correct for the given angle DEF

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44


Tier & Question Box plots3-5 4-6 5-7 6-8


a 1m 6

b 1m Gives a correct justifi cationeg

Median for year 10 = 56,Median for year 11 = 65,65 – 56 = 9

The medians are the vertical lines inside the

grey boxes, they are 412 divisions apart and

this is 9 marks since 1 division = 2 marks

Minimally acceptable justifi cationeg

56, 65 The medians are the vertical lines inside the boxes and they are 9 marks apart

There is a gap of 9 [with both medians indicated on the graph]

! Ambiguous notationeg

56 – 65Condone

Incomplete justifi cationeg

The difference between the medians is9 marks on the graph

c 1m

U1

Indicates Yesandgives a correct explanation, referring to the inter-quartile rangeeg

Inter-quartile range for year 10 = 33,Inter-quartile range for year 11 = 18, so year 11 was more consistent

The middle half of the year group was less spread out for year 11 than for year 10

The grey box shows the inter-quartile range and it is shorter for year 11


33, 18 Its inter-quartile range is 15 less The IQ range is smaller The IQ range is bigger for year 10 The box is shorter (or smaller) For Y10: 43 to 76, for Y11: 51 to 69 It is shorter [distance between upper and lower quartiles indicated on both box plots]

! ‘Inter-quartile range’ referred to as ‘range’ within an otherwise correct explanationAccept only if it is clear the response actually refers to the inter-quartile rangeeg, accept

For year 10, range = 33For year 11, range = 18

eg, do not accept The range is bigger for year 10


Year 11 is shorter than year 10 The results for year 10 are more spread out than in year 11

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45


Tier & Question Circle graph3-5 4-6 5-7 6-8


a 2m

or

Completes both pairs of coordinates correctly, ie

(3, 4) and (3, –4), in either order

1m Completes either pair of coordinates correctly

or

Shows the value 16

or

Shows or implies a correct method for fi nding the value of yeg

y2 = 25 – 32

b 1m 5 –5 or ± 5

c 2m

or

Gives P as (3.5, 3.5) ! For 2m, gives P as (–3.5, –3.5)Condone

For 2m, equivalent fractions or decimals

1m Shows the value 3.5(…) or 12.5 or equivalent

or

Shows or implies a correct method for fi nding the value of x or yeg

2y2 = 25 x2 = 25 ÷ 2

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46


Tier & Question Giant pandas3-5 4-6 5-7 6-8


2m

or

1100 ! For 2m upper bound usedSince pupils could assume 1600 is given to the nearest 100 in the context of the question, accept use of upper bound provided a correct method is seeneg, for 2m accept

1650 ÷ 140 × 100, answer: 1200

1m Shows the digits 11(…)

or

Shows or implies a complete correct methodeg

1600 ÷ 140 × 100

16001.4

160 000140

For 1m, lower and/or upper bound used within a correct methodeg, for 1m accept

1650 ÷ 140 × 100 1550 ÷ 1.4

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47


Tier & Question Prism3-5 4-6 5-7 6-8


3m

or

6.9(…), or 4√3, or √48 ! Value of 7Do not accept unless a correct method or a more accurate value is seen

For 3m, 2m or 1m, method uses accurate or scale drawing

2m

or

Shows or implies a correct method with not more than one computational or rounding erroreg

√32 + 16 √32 = 5.6 (rounding error)

AC2 = 5.62 + 42

AC = 6.8(…) √32 = 6 (premature rounding)

√36 + 16 = 7.2

1m

U1

Shows suffi cient working to indicate correct application of Pythagoras’ theorem for at least one triangle eg

42 + 42

2 × 16 5.6(…) or 5.7 seen (Their BC)2 + 42

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48


Tier & Question Number cards3-5 4-6 5-7 6-8


2m

or

Gives all three correct values, ie

1520

25 in any order

1m

U1

Gives any two correct values, with not more than one error or omission

or

States or implies that n is a multiple of 5 and that

there are n5

square numbers

eg There must be 1 out of 5, 2 out of 10,

3 out of 15 etc for the fraction to be right 1 2 3 4 5, but should be only one

6 7 8 9 10, but should be only two

11 12 13 14 15, correct

! For 1m, minimally acceptable implicationFor 1m, accept responses in which there are at least three examples using multiples of 5, (with no examples not using multiples of 5) and some square numbers identifi ed, even if there are errors or omissionseg

1, 2, 3, 4, 5, so n could be 5

6, 7, 8, 9, 10, so n could be 10

11, 12, 13, 14, 15

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49


Tier & Question Window3-5 4-6 5-7 6-8


3m

or

Gives an integer value between 3925 and3928 inclusive

2m

or

Shows a non-integer value between 3925 and 3927.5 inclusive[rounding to the nearest whole number omitted]

or

Shows an integer value between 7850 and 7855 inclusive[division of whole circle area by 2 omitted]

or

Shows or implies a complete correct method with not more than one error, and follows through to give their value correct to the nearest whole numbereg

1m ÷ 2 = 50cm,

π × 502

2 = integer response outside

correct range π × 0.5 × 0.5 = 0.79 (premature rounding),

0.79 ÷ 2 = 0.395,0.395 × 10000 = 3950

π × 0.52

2 × 100 (error) = 39

For 2m or 1m, conceptual erroreg

π × 100 ÷ 2 = 157

For 2m uses a radius of 25 or 0.25

1m

U1

Shows a non-integer value between 7850 and 7855 inclusive

or

Shows the value 0.39(…) or equivalent[ie, the correct area in m²]

or

Shows or implies a complete correct method with not more than one error but fails to follow through to give their value correct tothe nearest whole numbereg

1m ÷ 2 = 50cm,

π × 502

2 = non-integer response outside

correct range

π × 252 (error) ÷ 2 = 981.75

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50

2008 KS3 Mathematics test mark scheme: Paper 2 Index

Tier Question Page

3–5 4–6 5–7 6–8

1 Rounding 11

2 Cuboid 12

3 Placing 40 12

4 Directions 13

5 Writing cheques 14

6 Theme park 14

7 Adding odd 15

8 Calculating 15

9 1 Time machine 16

10 2 Four cards 16

11 3 Sleep 17

12 4 Sorting shapes 17

13 5 Shopping 18

14 6 Speedometer 18

15 7 Football survey 19

16 8 Jug 19

17 9 Double shape 20

18 10 1 Cube edges 22

19 11 2 Track 22

20 12 3 Matching expressions 23

21 13 4 Area 23

22 14 5 Values 23

23 15 6 Symmetry patterns 24

24 17 7 Shop 25

25 18 8 Using algebra 26

26 16 9 Goldbach 27

27 19 10 Side length 28

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51

2008 KS3 Mathematics test mark scheme: Paper 2 Index

Tier Question Page

3–5 4–6 5–7 6–8

20 11 1 Value of x 29

21 12 2 Darts 29

22 13 3 Conversions 30

23 14 4 Concorde 31

24 15 5 Counters in a bag 32

25 16 6 Perimeters 33

26 17 7 Yoghurt 33

27 18 8 Lawn 34

28 19 9 Triangular numbers 34

29 21 10 Isosceles triangle 35

30 20 11 Journeys 36

22 12 Special offer 37

23 13 Planes 38

24 14 Cubes 39

25 15 Best buy 40

26 16 Shadows 41

27 17 1, 2, 4 42

28 18 Triangles 43

19 Box plots 44

20 Circle graph 45

21 Giant pandas 46

22 Prism 47

23 Number cards 48

24 Window 49

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First published 2008

© Qualifications and Curriculum Authority 2008

ISBN 1-84721-491-6

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